Hoheisel, Tim; Pablos, Blanca; Pooladian, Aram; Schwartz, Alexandra; Steverango, Luke A study of one-parameter regularization methods for mathematical programs with vanishing constraints. (English) Zbl 1517.65047 Optim. Methods Softw. 37, No. 2, 503-545 (2022). Summary: Mathematical programs with vanishing constraints (MPVCs) are a class of nonlinear optimization problems with applications to various engineering problems such as truss topology design and robot motion planning. MPVCs are difficult problems from both a theoretical and numerical perspective: the combinatorial nature of the vanishing constraints often prevents standard constraint qualifications and optimality conditions from being attained; moreover, the feasible set is inherently nonconvex, and often has no interior around points of interest. In this paper, we therefore study and compare four regularization methods for the numerical solution of MPVCs. Each method depends on a single regularization parameter, which is used to embed the original MPVC into a sequence of standard nonlinear programs. Convergence results for these methods based on both exact and approximate stationary of the subproblems are established under weak assumptions. The improved regularity of the subproblems is studied by providing sufficient conditions for the existence of KKT multipliers. Numerical experiments, based on applications in truss topology design and an optimal control problem from aerothermodynamics, complement the theoretical analysis and comparison of the regularization methods. The computational results highlight the benefit of using regularization over applying a standard solver directly, and they allow us to identify two promising regularization schemes. MSC: 65K05 Numerical mathematical programming methods 90C30 Nonlinear programming 90C31 Sensitivity, stability, parametric optimization 49J15 Existence theories for optimal control problems involving ordinary differential equations 49M20 Numerical methods of relaxation type Keywords:mathematical program with vanishing constraints; constraint qualification; optimality conditions; M-stationarity; T-stationarity; regularization method; truss topology optimization; aerothermodynamics PDFBibTeX XMLCite \textit{T. Hoheisel} et al., Optim. Methods Softw. 37, No. 2, 503--545 (2022; Zbl 1517.65047) Full Text: DOI arXiv References: [1] Achtziger, W.; Hoheisel, T.; Kanzow, C., On a relaxation method for mathematical programs with vanishing constraints, GAMM-Mitteilungen, 35, 110-130 (2012) · Zbl 1256.49032 [2] Achtziger, W.; Hoheisel, T.; Kanzow, C., A smoothing-regularization approach to mathematical programs with vanishing constraints, Comput. Optim. 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